Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
17,801 questions
14
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Was Fermat's Last Theorem known for infinitely many primes before Wiles?
Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$.
Is there any infinite class of primes ...
- 1,125
2
votes
1
answer
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Identity using $\varphi(n), d(n)$ and $\sigma(n)$
Let
$\varphi(n)$ be the Euler totient function.
$d(n)$ be the number of divisors of $n$.
$\sigma(n)$ be the sum of the divisors of $n$.
$a(n)$ be A344598, i.e., an integer sequence such that $$ a(n) =...
-6
votes
0
answers
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Is this proof that $A^3+B^3=C^n$ has no primitive solutions correct? [closed]
I am an independent researcher. This arose in the context of studying the Beal conjecture.
Setup: Factor $A^3+B^3=(A+B)(A^2-AB+B^2)=C^n$. For coprime $A,B$: $\gcd(A+B, A^2-AB+B^2)$ divides $3$. This ...
6
votes
0
answers
202
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Composite number $n$ with most $k \le n$ such that $n \mid \binom nk$
This problem arised in a local forum, proposed by a user named zxt.
Let $f(n)$ be the number of nonnegative integer $k$ not greater than $n$ such that $n \mid \binom{n}{k}$. If for each positive ...
- 221
0
votes
0
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165
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Finding all integer solutions to a family of elliptic curves depending on a parameter $n$
Consider this equation
\begin{equation}
y^2 = x^3 + (36n + 27)^2 \cdot x^2 + (15552 n^3 + 34992 n^2 + 26244 n + 6561) \cdot x + (46656 n^4 + 139968 n^3 + 157464 n^2 + 78713 n + 14748)
\end{equation}
...
- 139
13
votes
1
answer
475
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Find the gcd of $n^{2} + 1$ and $n! + 1$
I want to find the $\gcd$ of $n^{2} + 1$ and $n! + 1$. I have verified the first $10,000$ $n$ using a Python program. Their results are all $1$. So is it true that for all $n \geq 2$ , $n^2 + 1$ ...
- 131
4
votes
1
answer
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Is the $\Xi(n) = \prod_{k=1}^{\infty} \text{lcm}(1, 2, \dots, \lfloor n^{1/k} \rfloor)$ sequence a subset of the highly abundant numbers?
In this previous discussion, it was demonstrated that the standard Least Common Multiple sequence $\text{lcm}(1, 2, \dots, n)$ is not a subset of the highly abundant numbers.
In analytic number theory,...
- 1,287
2
votes
0
answers
80
views
Fallback for failure case in Galois factoring with units
Let $N = x^2 + 3y^2$ be a composite integer with a known representation of this form. Consider the cubic polynomial
$$
f(t) = 4t^3 - 3Nt - Nx,
$$
and let $K = \mathbb{Q}(\alpha)$ be the cubic number ...
- 427
4
votes
1
answer
150
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+50
What uniform Selberg–Delange estimate is needed to justify this contour shift?
In a very helpful answer by Thurmond (A weighted sum over squarefree numbers involving Bernoulli numbers), Thormund reduces the problem to controlling the integral on the shifted line $\Re z=-3/2$, ...
- 826
1
vote
0
answers
131
views
On smooth pronic numbers
By Størmer's theorem, for every fixed $k$, there are only finitely many $n$ such that the set of prime divisors of $n(n+1)$ is a subset of the first $k$ primes. The OEIS sequence A141399 tracks those $...
- 1,125
4
votes
1
answer
237
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Does this specific polynomial identity generate infinitely many triples satisfying $c > \text{rad}(abc)$?
Starting from a math problem involving a square and a unit circle, I used some elementary algebraic transformations and discovered that for any arbitrary $u, v$, if we define $a, b, c$ as follows:
$$a ...
- 2,452
4
votes
1
answer
226
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Are the highly abundant "pure factorial products" numbers $\Xi_{0} = \prod_{k=2}^{\infty} k!^{a_k}$ infinite?
While investigating the structural properties of highly abundant numbers (HA), I analyzed their decomposition into a "factorial-base" and then I proposed the two following definitions:
...
- 1,287
6
votes
1
answer
288
views
Elegant identity using Stirling numbers of both kinds and Bernoulli numbers
Let
$B_n$ be the $n$-th Bernoulli number with $B_1 = -\frac{1}{2}$.
$T_m(n,k)$ be the family of integer coefficients such that $$T_m(n,k) = \sum\limits_{j=k}^{n} {n \brace j} {j \brack k} j^m. $$
$...
user587997
2
votes
0
answers
108
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Whether consecutive maximal prime gaps exist for $p > 3$?
Question: Among the 84 known maximal prime gaps, consecutive records only occur for $p=2$ and $p=3$. Is it possible that for all $p > 3$, record-breaking gaps are strictly isolated?
Inquiry: I am ...
- 2,452
-3
votes
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answers
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Does the self-consistency constraint on hypothetical off-line zeta zeros improve the de Bruijn–Newman bound? [closed]
Consider a hypothetical zero $\rho^* = \tfrac{1}{2} + h_0 + it_0$ of $\zeta(s)$ with $h_0 > 0$. From the partial-fraction expansion of $\zeta'/\zeta$, the second derivative of $f(h) = \log|\xi(\...
